Axiom ax-mulass 7916

Description: Multiplication of complex numbers is associative. Axiom for real and complex numbers, justified by Theorem axmulass 7874. Proofs should normally use mulass 7944 instead. (New usage is discouraged.) (Contributed by NM, 22-Nov-1994.)

Assertion

Ref	Expression
ax-mulass	⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 · 𝐵) · 𝐶) = (𝐴 · (𝐵 · 𝐶)))

Detailed syntax breakdown of Axiom ax-mulass

Step	Hyp	Ref	Expression
1		cA	. . . 4 class 𝐴
2		cc 7811	. . . 4 class ℂ
3	1, 2	wcel 2148	. . 3 wff 𝐴 ∈ ℂ
4		cB	. . . 4 class 𝐵
5	4, 2	wcel 2148	. . 3 wff 𝐵 ∈ ℂ
6		cC	. . . 4 class 𝐶
7	6, 2	wcel 2148	. . 3 wff 𝐶 ∈ ℂ
8	3, 5, 7	w3a 978	. 2 wff (𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ)
9		cmul 7818	. . . . 5 class ·
10	1, 4, 9	co 5877	. . . 4 class (𝐴 · 𝐵)
11	10, 6, 9	co 5877	. . 3 class ((𝐴 · 𝐵) · 𝐶)
12	4, 6, 9	co 5877	. . . 4 class (𝐵 · 𝐶)
13	1, 12, 9	co 5877	. . 3 class (𝐴 · (𝐵 · 𝐶))
14	11, 13	wceq 1353	. 2 wff ((𝐴 · 𝐵) · 𝐶) = (𝐴 · (𝐵 · 𝐶))
15	8, 14	wi 4	1 wff ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 · 𝐵) · 𝐶) = (𝐴 · (𝐵 · 𝐶)))

This axiom is referenced by:  mulass  7944